Abstract:
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Maximum likelihood continues to be a theme in current statistical theory in both parametric and nonparametric settings despite a number of known potential difficulties: - Maximum likelihood estimators may not exist. - When MLE's exist, they may not be consistent. - When MLE's exist and are consistent, they may not attain minimax rates of convergence. In spite of these difficulties, maximum likelihood has also had a number of success stories in semiparametric and nonparametric problems. The talk will survey some of the difficulties and a selection from recent progress, including: (a) the ugly: non-existence, non-uniqueness, and inconsistency. (b) the bad: possible non-attainment of minimax rates in high-dimensional (trans-Donsker) settings (c) the good: progress on (1) beyond consistency for Kiefer-Wolfowitz mixture models; (2) behavior of profile-likelihood methods for semiparametric models; (3) behavior of shape constrained estimators globally, locally, and under model-misspecification.
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